# In this investigation, I will attempt to find out some of the properties of a 2x2 square drawn within a 10x10 number grid. After this I hope to be able to investigate 9x9 and 8x8 grids, next I hope to move on to investigating rectangles.

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Introduction

Daniel Lovegrove 10CW

Maths Coursework

MATHS

NumberGridCoursework

## Introduction

In this investigation, I will attempt to find out some of the properties of a 2x2 square drawn within a 10x10 number grid. After this I hope to be able to investigate 9x9 and 8x8 grids, next I hope to move on to investigating rectangles.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |

21 | 22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 |

31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 |

41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |

51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 |

61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 |

71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 | 79 | 80 |

81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 |

91 | 92 | 93 | 94 | 95 | 96 | 96 | 97 | 98 | 100 |

such as

35 | 36 |

45 | 46 |

when I find the product of the 2 sets of opposite corners I also find that the difference between these answers is ten:

2x2 squares square products difference | |||||||||||

| 45 x 36 = 1620 35 x 46 = 1610 | 10 | |||||||||

| 57 x 68 = 3876 67 x 58 = 3886 | 10 | |||||||||

| 85 x 96 = 8160 95 x 86 = 8170 | 10 |

My findings are laid out here algebraically (s= starting number)

35 | 36 |

45 | 46 |

The +10 in the lower left corner is there because every row has ten numbers and so if you look down the columns you will see that

here 35+10=40 and 36+10=46 and the

+1 is there obviously because every

number you count to the right is bigger

by 1 i.e. 35+1=36 I shall use the same method in all my working throughout this project

S | S+1 |

S+10 | S+11 |

S(S+11) = S2 +11S

(S+10)(S+1) = S2 +10S +S + 10 = S2 + 11S + 10

The difference is ten

To find this I multiplied the corners of my square and found the difference between the products.

Next I investigated 3 x 3 squares within a 10 x 10 grid and found that the difference was 40

Square products difference | |||||||||||||

| 1 x 23 = 23 21 x 3 = 63 | 40 | |||||||||||

| 55 x 77 = 4235 75 x 57 = 4275 | 40 | |||||||||||

| 44 x 66 = 2904 64 x 46 = 2944 | 40 |

Again my results are laid out algebraically (s = starting number)

S | S+1 | S+2 |

S+10 | S+11 | S+12 |

S+20 | S+21 | S+22 |

S(S+22)

Middle

S(S+10) = S2+10S

(S+9)(S+1) = S2+10s+9

The difference is nine

My investigation continues with 3x3 squares

Square | Products | Difference | |||||||||||||

| 29x49=1421 47x31=1457 | 36 | |||||||||||||

| 13x33=429 31x15=465 | 36 | |||||||||||||

| 48x68=3264 66x50=3300 | 36 |

S | S+1 | S+2 |

S+9 | S+10 | S+11 |

S+18 | S+19 | S+20 |

S(S+20) = S2+20S

(S+18)(S+2) = S2+20S+36

The difference is 36

Now I am investigating 4x4 squares

Square | Products | Difference | |||||||||||||||||

| 30x60=1800 57x33=1881 | 81 | |||||||||||||||||

| 4x34=136 31x7=217 | 81 | |||||||||||||||||

| 22x52=1144 49x25=1225 | 81 |

S | S+1 | S+2 | S+3 |

S+9 | S+10 | S+11 | S+12 |

S+18 | S+19 | S+20 | S+21 |

S+27 | S+28 | S+29 | S+30 |

S(S+30) = S2+30S

(S+27)(S+3) = S2+30S+81

the difference is 81

My second theory (see Above†) will be proved right or wrong by my investigations into 5x5 squares within a 9x9 grid.

Square | Products | Difference | ||||||||||||||||||||||||||

| 12x52=624 48x16=768 | 144 | ||||||||||||||||||||||||||

| 41x81=3321 77x45=3465 | 144 | ||||||||||||||||||||||||||

| 28x68=1904 64x32=2048 | 144 |

S | S+1 | S+2 | S+3 | S+4 |

S+9 | S+10 | S+11 | S+12 | S+13 |

S+18 | S+19 | S+20 | S+21 | S+22 |

S+27 | S+28 | S+29 | S+30 | S+31 |

S+36 | S+37 | S+38 | S+39 | S+40 |

S(S+40) = S2+S40

(S+36)(S+4) = S2+S40+144

This is the correct difference

My second theory (see Above†) was correct

I am now going to produce a formula for squares in a 9x9 grid

1 | 2 |

10 | 11 |

As before, I shall use my algebraic values to investigate 2x2, 3x3, 4x4 and 5x5 squares

W

S | S+(w-1) |

S+(9[w-1]) | S+(9[w-1])+(w-1) |

S(S+9W-9+W-1) = 1(11) =11

(S+9W-9)(S+W-1) = (10)(2)=20

the difference is 9, which is what it should be

29 | 30 | 31 |

38 | 39 | 40 |

47 | 48 | 49 |

S | S+(W-1) | |

S+9(W-1) | S+9(W-1) +(W-1) |

S(S+9[W-1]+[W-1]) = 29(49) = 1421

(S+9[W-1])(S+[W-1] = (47)(31) = 1457

the difference is 36, which is the correct difference

22 | 23 | 24 | 25 |

31 | 32 | 33 | 34 |

40 | 41 | 42 | 43 |

49 | 50 | 51 | 52 |

Conclusion

(4X5)

Rectangle | Products | Difference | ||||||||||||||||||||||||

| 51X94=4794 91X54=4914 | 120 | ||||||||||||||||||||||||

| 6X49=294 46X9=414 | 120 | ||||||||||||||||||||||||

| 1X44=44 41X4=164 | 120 |

If I modify my formula for the difference of any size square in a 10X10 grid, I can create a formula for any size rectangle in a 10X10 grid.

I already have the terms:

- S=the starting number
- W=the width

Now I need one for the length I shall use O.

24 | 25 |

34 | 35 |

44 | 45 |

To test my formula I shall use the rectangle

For this rectangle

O=3

W=2

S=24

S | S+(W-1) |

S+10(O-1) | S+10(O-1)+(W-1) |

S(S+10[O-1]+[W-1]) = 24(45) = 1080

(S+10[O-1])(S+[W-1]) = (44)(25) = 1100

the difference is 20 which is the correct number for this rectangle

Page of

[1]* I now have a theory ; I think that the number in the difference column in a table for a 2x2 grid is the same as the size of the grid the 2x2 square is in, I shall investigate this further after I have completed my investigation of a 9x9 grid

[2]† I have another theory, I think that the numbers in the difference column for a square in a 9x9 grid are 90% of the difference of the corresponding size squares in a 10x10 grid just as the size of a 9x9 grid is 90% of the size of a 10x10

[3]‡ I am extending my second theory to say that the number in the difference column of a certain size square in an 8x8 grid is 80% of the same number for the same size square in a 10x10 grid and 70% for a 7x7 grid etc.

This student written piece of work is one of many that can be found in our GCSE Number Stairs, Grids and Sequences section.

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